Incomplete oblique projections for solving large inconsistent linear systems

Scolnik, Hugo D.; Echebest, Nélida Ester; Guardarucci, M. T.; Vacchino, M. C.

doi:10.1007/s10107-006-0066-4

Cited by 18 publications

(54 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…For instance, the oblique projections used by Arioli et al [1], Scolnik et al [30], Censor and Elfving [7] and Censor et al [9] and the projection mapping associated with the 1-norm, +∞-norm and p-norm onto a hyperplane used by Mangasarian [19,20]. For applications using gauge distances; that is, distances associated with gauge functions (see Remark 6), as well as distances associated with different norms see [6,27].…”

Section: Remarkmentioning

confidence: 97%

“…This view point allows the study of the notion of projection in several different contexts. Works dealing with projections associated to different distances include Carrizosa and Plastria [6], Censor and Elfving [7], Censor et al [9], Mangasarian [19,20], Dax [11,12], Plastria and Carrizosa [27] and Scolnik et al [30].…”

mentioning

confidence: 99%

“…[15]), Optimization (see e.g. [2,8,9,26,28,30,32]), Numerical Linear Algebra (see e.g. [31]), Statistics (see e.g.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Generalized projections onto convex sets

Ferreira¹,

Németh

2011

J Glob Optim

View full text Add to dashboard Cite

This paper introduces the notion of projection onto a closed convex set associated with a convex function. Several properties of the usual projection are extended to this setting. In particular, a generalization of Moreau's decomposition theorem about projecting onto closed convex cones is given. Several examples of distances and the corresponding generalized projections associated to particular convex functions are presented.

show abstract

Section: Remarkmentioning

confidence: 97%

mentioning

confidence: 99%

See 1 more Smart Citation

Generalized projections onto convex sets

Ferreira¹,

Németh

2011

J Glob Optim

View full text Add to dashboard Cite

show abstract

“…The aim of this paper is to apply a convex regularizing function for improving the performance of the incomplete oblique projections (IOP) method, introduced in Scolnik et al (2008b) by the authors for solving linear least squares problems, when applied to image reconstruction problems. That method uses a scheme of IOP onto the solution set of the augmented system Ax − r = b , and converges to a weighted least squares solution of the system Ax = b when rank ( A )= n .…”

Section: Introductionmentioning

confidence: 99%

“…In order to solve , we have defined two convex sets in the (2 n + m )‐dimensional space , denoted by [ u ; v ] the vertical concatenation of , with , and for all , D is a diagonal matrix of order 2 n + m , whose n first elements are 1's, the next m coincide with those of D m , and the last n elements are those of D n . By means of a direct application of the Karush–Kuhn–Tucker (KKT) conditions (Luenberger, 1986) to the problem it is possible to prove (see Scolnik et al, 2008b) that this is equivalent to . That observation led us to use the IOP algorithm for solving , applying projections scheme between the sets and , as in the original development in Scolnik et al (2008b), replacing the computation of the exact projections onto by suitable incomplete or approximate projections (Scolnik et al, 2008a).…”

Section: Introductionmentioning

confidence: 99%

Implicit regularization of the incomplete oblique projections method

Scolnik¹,

Echebest²,

Guardarucci³

2009

Int Trans Operational Res

Self Cite

View full text Add to dashboard Cite

The aim of this paper is to improve the performance of the incomplete oblique projections method (IOP), previously introduced by the authors for solving inconsistent linear systems, when applied to image reconstruction problems. That method employs incomplete oblique projections onto the set of solutions of the augmented system Ax−r=b, and converges to a weighted least squares solution of the system Ax=b. Many tomographic image reconstruction problems are such that the limitation of the range of rays makes the model underdetermined, the discretized linear system is rank‐deficient, the nullspace is non‐trivial, and the minimal norm least squares solution may be far away from the true image. In a previous paper, we have added a quadratic term reflecting neighboring pixel information to the standard least squares model for improving the quality of the reconstructed images. In this paper we replace the quadratic function by a more general regularizing function avoiding the modification of the original system. The key idea is to perform a joint optimization of the norm of the residual and of the regularizing function in each iteration. The theoretical properties of this new algorithm are analyzed, and numerical experiments are presented comparing its performance with other well‐known methods. They show that the new approach improves the quality of the reconstructed images.

show abstract

Introduction

Cegielski

2012

Lecture Notes in Mathematics

View full text Add to dashboard Cite

Incomplete oblique projections for solving large inconsistent linear systems

Cited by 18 publications

References 14 publications

Generalized projections onto convex sets

Generalized projections onto convex sets

Implicit regularization of the incomplete oblique projections method

Introduction

Contact Info

Product

Resources

About